Regularity of the leafwise Poincare metric on singular holomorphic foliations

Let $\mathcal F$ be a smooth Riemann surface foliation on $M \setminus E$, where $M$ is a complex manifold and the singular set $E \subset M$ is an analytic set of codimension at least two. Fix a hermitian metric on $M$ and assume that all leaves of $\mathcal F$ are hyperbolic. Verjovsky's modulus of uniformization $\eta$ is a positive real function defined on $M \setminus E$ defined in terms of the family of holomorphic maps from the unit disc $\mathbb D$ into the leaves of $\mathcal F$ and is a measure of the largest possible derivative in the class of such maps. Various conditions are known that guarantee the continuity of $\eta$ on $M \setminus E$. The main question that is addressed here is its continuity at points of $E$. To do this, we adapt Whitney's $C_4$-tangent cone construction for analytic sets to the setting of foliations and use it to define the tangent cone of $\mathcal F$ at points of $E$. This leads to the definition of a foliation that is of {\it transversal type} at points of $E$. It is shown that the map $\eta$ associated to such foliations is continuous at $E$ provided that it is continuous on $M \setminus E$ and $\mathcal F$ is of transversal type. We also present observations on the locus of discontinuity of $\eta$. Finally, for a domain $U \subset M$, we consider $\mathcal F_U$, the restriction of $\mathcal F$ to $U$ and the corresponding positive function $\eta_U$. Using the transversality hypothesis leads to strengthened versions of the results of Lins Neto--Martins on the variation $U \mapsto \eta_U$.